Coresets with arbitrary weight functions (i.e. with negative weights allowed) have been considered [FL11, etc]. However, computing approximate solutions on these coresets in polynomial-time remains a challenge, so we restrict our definition to non-negative weight functions. This ensures that an approximate solution can be quickly produced. This implies a PTAS for Euclidean space and a polynomial-time \(2\gamma (1+\epsilon )\) -approximation for general metric spaces (where \(\gamma \)  is the best polynomial-time approximation factor for the problem in the batch setting). This factor of \(2\gamma (1+\epsilon )\)  is well-known in the literature, see [1]}, [2]}, [3]} for details.
